C HAPTER

8

Similarity

Chapter Outline 8.1

R ATIOS AND P ROPORTIONS

8.2

S IMILAR P OLYGONS

8.3

S IMILARITY BY AA

8.4

S IMILARITY BY SSS AND SAS

8.5

P ROPORTIONALITY R ELATIONSHIPS

8.6

S IMILARITY T RANSFORMATIONS

8.7

E XTENSION : S ELF -S IMILARITY

8.8

C HAPTER 8 R EVIEW

In this chapter, we will start with a review of ratios and proportions. Second, we will introduce the concept of similarity. Two ﬁgures are similar if they have the same shape, but not the same size. We will apply similarity to polygons, quadrilaterals and triangles. Then, we will extend this concept to proportionality with parallel lines and dilations. Finally, there is an extension about self-similarity, or fractals, at the end of the chapter.

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Chapter 8. Similarity

8.1 Ratios and Proportions Learning Objectives

• Write, simplify, and solve ratios and proportions. • Use ratios and proportions in problem solving. Review Queue

1. Are the two triangles congruent? If so, how do you know?

2. If AC = 5, what is GI? What is the reason? 3. How many inches are in a foot? In a yard? In 3 yards? 4. How many cups are in a pint? In a quart? In a gallon? In 7 quarts? Know What? You want to make a scale drawing of your room and furniture for a little redecorating. Your room measures 12 feet by 12 feet. Also in your room is a twin bed (36 in by 75 in), a desk (4 feet by 2 feet), and a chair (3 feet by 3 feet). You decide to scale down your room to 8 in by 8 in, so the drawing ﬁts on a piece of paper. What size should the bed, desk and chair be? Draw an appropriate layout for the furniture within the room. Do not round your answers. Using Ratios

Ratio: A comparison of two numbers. Ratios can be written: ab , a : b, and a to b. Example 1: The total bagel sales at a bagel shop for Monday is in the table below. What is the ratio of cinnamon raisin bagels to plain bagels?

TABLE 8.1: Type of Bagel Plain Cinnamon Raisin Sesame Jalapeno Cheddar Everything Honey Wheat

Number Sold 80 30 25 20 45 50

Solution: The ratio is 30 80 , 30:80, or 30 to 80. Depending on the problem, ratios are usually written in simplest form, which means to reduce the ratio. The answer is then 38 , 3:8, or 3 to 8. 411

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Example 2: What is the ratio, in simplest form, of Honey Wheat bagels to total bagels sold? Solution: Remember that order matters. Because the Honey Wheat is listed ﬁrst, that is the number that comes ﬁrst in the ratio (on in the numerator of the fraction). Find the total number of bagels sold.

80 + 30 + 25 + 20 + 45 + 50 = 250 The ratio is then

50 250

= 15 , 1:5, or 1 to 5.

We call the ratios 50:250 and 1:5 equivalent because one reduces to the other. In some problems you may need to write a ratio of more than two numbers. For example, the ratio of the number of cinnamon raisin bagels to sesame bagels to jalapeno cheddar bagels is 30:25:20 or 6:5:4. This is known as an extended ratio. Measurements are used a great deal with ratios and proportions. For example, how many feet are in 2 miles? How many inches are in 4 feet? Students are expected to know basic conversion factors. Example 3: Simplify the following ratios. a)

7 ft 14 in

b) 9m : 900cm c)

4 gal 16 gal

Solution: Change these so that they are in the same units. [email protected] f t 12 in 84 6 = 14 = 1 a) @ · ft 14 in 1 @ @ Notice that the inches cancel each other out. Ratios should not contain any units once simpliﬁed.

b) It is easier to simplify ratios when they are written as fractions. c)

4 gal 16 gal

=

9m 900 cm

cm · 100 1m =

900 900

=

1 1

1 4

Example 4: A talent show features dancers and singers. The ratio of dancers to singers is 3:2. There are 30 performers total, how many singers are there? Solution: 3:2 is a reduced ratio, so there is a whole number, n, that we can multiply both by to ﬁnd the total number in each group.

dancers = 3n, singers = 2n −→ 3n + 2n = 30 5n = 30 n=6 Therefore, there are 3 · 6 = 18 dancers and 2 · 6 = 12 singers. To double-check, 18 + 12 = 30 total performers. Proportions

Proportion: Two ratios that are equal to each other. Example 4: Solve the proportions. a) b)

4 x 5 = 30 y+1 5 8 = 20

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6 5

=

Chapter 8. Similarity

2x+5 x−2

Solution: To solve a proportion, cross-multiply. a)

b)

c)

In proportions, the blue numbers are called the means and the orange numbers are called the extremes. For the proportion to be true, the product of the means must equal the product of the extremes. This can be generalized in the Cross-Multiplication Theorem. Cross-Multiplication Theorem: Let a, b, c, and d be real numbers, with b = 0 and d = 0. If

a b

= dc , then ad = bc.

The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by 22 , bb , or because it is the same number divided by itself (b ÷ b = 1).

d d

=1

Proof of the Cross-Multiplication Theorem

a c = b d c b a d · = · b d d b ad bc = bd bd ad = bc

Multiply the left side by

d b and the right side by . d b

The denominators are the same, so the numerators are equal.

Think of the Cross-Multiplication Theorem as a shortcut. Without this theorem, you would have to go through all of these steps every time to solve a proportion. Example 5: Your parents have an architect’s drawing of their home. On the paper, the house’s dimensions are 36 in by 30 in. If the shorter length of your parents’ house is actually 50 feet, what is its actual longer length? 413

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Solution: Set up a proportion. If the shorter length is 50 feet, then it will line up with 30 in. It does not matter which numbers are in the numerators of the fractions, as long as they line up correctly. 30 50 = −→ 1800 = 30x 36 x 60 = x So, the dimension of your parents’ house is 50 ft by 60 ft. Properties of Proportions

The Cross-Multiplication Theorem has several sub-theorems that follow from its proof. The formal term forthis is a corollary. Corollary: A theorem that follows quickly, easily, and directly from another theorem. Below are three corollaries that are immediate results of the Cross Multiplication Theorem and the fundamental laws of algebra. Corollary 8-1: If a, b, c, and d are nonzero and

a b

= dc , then

a c

= db .

Corollary 8-2: If a, b, c, and d are nonzero and

a b

= dc , then

d b

= ac .

Corollary 8-3: If a, b, c, and d are nonzero and

a b

= dc , then

b a

= dc .

In other words, a true proportion is also true if you switch the means, switch the extremes, or ﬂip it upside down. Notice that you will still end up with ad = bc after cross-multiplying for all three of these corollaries. Example 6: Suppose we have the proportion this one.

2 5

=

14 35 .

Write down the other three true proportions that follow from

Solution: First of all, we know this is a true proportion because you would multiply three corollaries, we would get: 1. 2. 3.

2 5

by

7 7

to get

14 35 .

Using the

2 5 14 = 35 35 14 5 = 2 5 35 2 = 14

If you cross-multiply all four of these proportions, you would get 70 = 70 for each one. Corollary 8-4: If a, b, c, and d are nonzero and

a b

= dc , then

a+b b

=

c+d d .

Corollary 8-5: If a, b, c, and d are nonzero and

a b

= dc , then

a−b b

=

c−d d .

Example 7: In the picture,

AB XY

= YBCZ =

AC XZ .

Find AC and XY .

Solution: This is an example of an extended proportion. Substituting in the numbers for the sides we know, we have 4 3 AC XY = 9 = 15 . Separate this into two different proportions and solve for XY and AC. 414

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Chapter 8. Similarity

3 AC = 9 15 9(AC) = 45

4 3 = XY 9 36 = 3(XY ) XY = 12 Example 8: In the picture,

ED AD

=

BC AC .

AC = 5

Find y.

Solution: Substituting in numbers for the given sides:

6 8 = . −→ 8y = 6(20) y 12 + 8 y = 15 Example 9: If

AB BE

AC = CD in the picture above, ﬁnd BE.

Solution:

12 20 = −→ 20(BE) = 12(25) BE 25 BE = 15 Know What? Revisited Everything needs to be scaled down by a factor of into inches and then multiply by the scale factor.

1 18

(144 in ÷ 8 in). Change everything

Bed: 36 in by 75 in −→ 2 in by 4.167 in Desk: 48 in by 24 in −→ 2.67 in by 1.33 in Chair: 36 in by 36 in −→ 2 in by 2 in There are several layout options for these three pieces of furniture. Draw an 8 in by 8 in square and then the appropriate rectangles for the furniture. Then, cut out the rectangles and place inside the square. Review Questions

1. The votes for president in a club election were: Smith : 24 Munoz : 32 Park : 20 Find the following ratios and write in simplest form. (a) Votes for Munoz to Smith 415

8.1. Ratios and Proportions

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(b) Votes for Park to Munoz (c) Votes for Smith to total votes (d) Votes for Smith to Munoz to Park Use the picture to write the following ratios for questions 2-6.

AEFD is a square ABCD is a rectangle 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

AE : EF EB : AB DF : FC EF : BC Perimeter ABCD: Perimeter AEFD: Perimeter EBCF The measures of the angles of a triangle are have the ratio 3:3:4. What are the measures of the angles? The lengths of the sides in a triangle are in a 3:4:5 ratio. The perimeter of the triangle is 36. What are the lengths of the sides? The length and width of a rectangle are in a 3:5 ratio. The perimeter of the rectangle is 64. What are the length and width? The length and width of a rectangle are in a 4:7 ratio. The perimeter of the rectangle is 352. What are the length and width? The ratio of the short side to the long side in a parallelogram is 5:9. The perimeter of the parallelogram is 112. What are the lengths of the sides? The length and width of a rectangle are in a 3:11 ratio. The area of the rectangle is 528. What are the length and width of the rectangle? c+d Writing Explain why a+b b = d is a valid proportion. HINT: Cross-multiply and see if it equals ad = bc. a−b Writing Explain why b = c−d d is a valid proportion. HINT: Cross-multiply and see if it equals ad = bc.

Solve each proportion. x 15. 10 = 42 35 x 16. x−2 = 57 y 17. 69 = 24 18. 9x = 16 x y+6 19. y−3 = 8 5 20 20. z+5 = 16 7 21. Shawna drove 245 miles and used 8.2 gallons of gas. At the same rate, if she drove 416 miles, how many gallons of gas will she need? Round to the nearest thousandth. 22. The president, vice-president, and ﬁnancial ofﬁcer of a company divide the proﬁts is a 4:3:2 ratio. If the company made $1,800,000 last year, how much did each person receive?

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Chapter 8. Similarity

23. Many recipes describe ratios between ingredients. For example, one recipe for paper-mache paste suggests 3 parts ﬂour to 5 parts water. If we have one cup of ﬂour, how much water should we add to make the paste? 24. A recipe for krispy rice treats calls for 6 cups of rice cereal and 40 large marshmallows. You want to make a larger batch of goodies and have 9 cups of rice cereal. How many large marshmallows do you need? However, you only have the miniature marshmallows at your house. You ﬁnd a list of substitution quantities on the internet that suggests 10 large marshmallows are equivalent to 1 cup miniatures. How many cups of miniatures do you need? Given the true proportion, true. 25. 26. 27. 28.

=

15 d

=

x y

and d, x, and y are nonzero, determine if the following proportions are also

10 x y = 6 15 d 10 = 6 y+x 6+10 10 = x y 15 x = d

For questions 29-32,

29. 30. 31. 32.

10 6

AE ED

BC = CD and

ED AD

= CD DB =

EC AB .

Find DB. Find EC. Find CB. Find AD.

Review Queue Answers

1. 2. 3. 4.

Yes, they are congruent by SAS. GI = 5 by CPCTC 12 in = 1 ft, 36 in = 3 ft, 108 in = 3 yrds 2c = 1 pt, 4c = 1 qt, 16 c = 4 qt = 1 gal, 28c = 7 qt

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8.2. Similar Polygons

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8.2 Similar Polygons Learning Objectives

• Recognize similar polygons. • Identify corresponding angles and sides of similar polygons from a similarity statement. • Calculate and apply scale factors. Review Queue

1. Solve the proportions. (a) (b) (c)

6 x 4 7 5 8

= = =

10 15 2x+1 42 x−2 2x

2. In the picture,

AB XZ

=

BC XY

= YACZ .

(a) Find AB. (b) Find BC.

Know What? A baseball diamond is a square with 90 foot sides. A softball diamond is a square with 60 foot sides. Are the two diamonds similar? If so, what is the scale factor? Explain your answer. Similar Polygons

Similar Polygons: Two polygons that are the same shape, but not necessarily the same size. Think about similar polygons as an enlargement or reduction of the same shape. So, more speciﬁcally, similar polygons have the same number of sides, congruent corresponding angles, and proportional corresponding sides. The symbol ∼ is used to represent similar. Here are some examples:

These polygons are not similar: 418

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Chapter 8. Similarity

Example 1: Suppose ABC ∼ JKL. Based on the similarity statement, which angles are congruent and which sides are proportional? Solution: Just like a congruence statement, the congruent angles line up within the statement. So, A ∼ = J, B ∼ = K, AB BC AC L. The same is true of the proportional sides. We write the sides in a proportion, = = . and C ∼ = JK

KL

JL

Because of the corollaries we learned in the last section, the proportions in Example 1 could be written several AB JK different ways. For example, BC = KL . Make sure to line up the corresponding proportional sides. Example 2: MNPQ ∼ RSTU. What are the values of x, y and z?

Solution: In the similarity statement, M ∼ = R, so z = 115◦ . For x and y, set up a proportion. 18 x = 30 25 450 = 30x x = 15

18 15 = 30 y 450 = 18y y = 25

Speciﬁc types of triangles, quadrilaterals, and polygons will always be similar. For example, because all the angles and sides are congruent, all equilateral triangles are similar. For the same reason, all squares are similar. We can take this one step further and say that all regular polygons (with the same number of sides) are similar. Example 3: ABCD is a rectangle with length 12 and width 8. UVW X is a rectangle with length 24 and width 18. Are these two rectangles similar?

Solution: Draw a picture. First, all the corresponding angles need to be congruent. In rectangles, all the angles are 8 3 2 3 = 23 , 18 congruent, so this condition is satisﬁed. Now, lets see if the sides are proportional. 12 24 = 4 . 3 = 4 . This tells us that the sides are not in the same proportion, so the rectangles are not similar. We could have also set up the 1 8 4 1 4 proportion as 12 24 = 2 and 18 = 9 . 2 = 9 to end up with the same conclusion. 419

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Scale Factors

If two polygons are similar, the lengths of corresponding sides are proportional. If k is the length of a side in one polygon, and m is the length of the corresponding side in the other polygon, then the ratio mk is the scale factor relating the ﬁrst polygon to the second. Scale Factor: In similar polygons, the ratio of one side of a polygon to the corresponding side of the other. Example 5: ABCD ∼ AMNP. Find the scale factor and BC.

2 3 Solution: Line up the corresponding proportional sides. AB : AM, so the scale factor is 30 45 = 3 or 2 . Because BC is in the bigger rectangle, we will multiply 40 by 32 because it is greater than 1. BC = 32 (40) = 60.

Example 6: Find the perimeters of ABCD and AMNP. Then ﬁnd the ratio of the perimeters. Solution: Perimeter of ABCD = 60 + 45 + 60 + 45 = 210 Perimeter of AMNP = 40 + 30 + 40 + 30 = 140 The ratio of the perimeters is 140:210, which reduces to 2:3. Theorem 8-2: The ratio of the perimeters of two similar polygons is the same as the ratio of the sides. In addition the perimeter being in the same ratio as the sides, all parts of a polygon are in the same ratio as the sides. This includes diagonals, medians, midsegments, altitudes, and others. Example 7: ABC ∼ MNP. The perimeter of ABC is 150 and AB = 32 and MN = 48. Find the perimeter of MNP. Solution: From the similarity statement, AB and MN are corresponding sides. So, the scale factor is The perimeter of MNP is 23 (150) = 100.

32 48

=

2 3

or 32 .

Know What? Revisited All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball 3 diamond. This means all the sides are in a 90 60 = 2 ratio. All the angles in a square are congruent, all the angles in both diamonds are congruent. The two squares are similar and the scale factor is 32 .

Review Questions

Determine if the following statements are true or false. 1. All equilateral triangles are similar. 420

www.ck12.org 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Chapter 8. Similarity

All isosceles triangles are similar. All rectangles are similar. All rhombuses are similar. All squares are similar. All congruent polygons are similar. All similar polygons are congruent. All regular pentagons are similar. BIG ∼ HAT . List the congruent angles and proportions for the sides. If BI = 9 and HA = 15, ﬁnd the scale factor. If BG = 21, ﬁnd HT . If AT = 45, ﬁnd IG. Find the perimeter of BIG and HAT . What is the ratio of the perimeters?

Use the picture to the right to answer questions 14-18.

14. 15. 16. 17. 18.

Find m E and m Q. ABCDE ∼ QLMNP, ﬁnd the scale factor. Find BC. Find CD. Find NP.

Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.

19.

20. 421

8.2. Similar Polygons

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21.

22. 23. ABC ∼ DEF Solve for x and y.

24. QUAD ∼ KENT Find the perimeter of QUAD.

25. CAT ∼ DOG Solve for x and y. 422

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Chapter 8. Similarity

26. PENTA ∼ FIV ER Solve for x.

27. MNO ∼ XNY Solve for a and b.

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8.2. Similar Polygons

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28. Trapezoids HAV E ∼ KNOT Solve for x and y.

9 29. Two similar octagons have a scale factor of 11 . If the perimeter of the smaller octagon is 99 meters, what is the perimeter of the larger octagon? 30. Two right triangles are similar. The legs of one of the triangles are 5 and 12. The second right triangle has a hypotenuse of length 39. What is the scale factor between the two triangles? 31. What is the area of the smaller triangle in problem 30? What is the area of the larger triangle in problem 30? What is the ratio of the areas? How does it compare to the ratio of the lengths (or scale factor)? Recall that the area of a triangle is A = 12 bh.

Review Queue Answers

1.

2.

424

(a) (b) (c) (a) (b)

x=9 x = 11.5 x=8 AB = 16 BC = 14

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Chapter 8. Similarity

8.3 Similarity by AA Learning Objectives

• Determine whether triangles are similar. • Understand AA for similar triangles. • Solve problems involving similar triangles. Review Queue

1.

(a) Find the measures of x and y. (b) The two triangles are similar. Find w and z.

2. Use the true proportion

6 8

=

x 28

=

27 y to

answer the following questions.

(a) Find x and y. (b) Write another true proportion. 6+x (c) Is 28 8 = 12 true? If you solve for x, is it the same as in part a? Know What? George wants to measure the height of a ﬂagpole. He is 6 feet tall and his shadow is 10 feet long. At the same time of day, the shadow of the ﬂagpole was 85 feet long. How tall is the ﬂagpole?

Angles in Similar Triangles

The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles are congruent too. Because a triangle’s interior angle measures total 180◦ , the third angle in any triangle is 180◦ minus 425

8.3. Similarity by AA

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the sum of the other two angle measures. Investigate what happens when two different triangles have the same angle measures; use Investigation 4-4 (Constructing a Triangle using ASA) to help with this. Investigation 8-1: Constructing Similar Triangles Tools Needed: pencil, paper, protractor, ruler 1. Draw a 45◦ angle. Extend the horizontal side and then draw a 60◦ angle on the other side of this side. Extend the other side of the 45◦ angle and the 60◦ angle so that they intersect to form a triangle. What is the measure of the third angle? Measure the length of each side.

2. Repeat Step 1 and make the horizontal side between the 45◦ and 60◦ angle at least 1 inch longer than in Step 1. This will make the entire triangle larger. Find the measure of the third angle and measure the length of each side.

3. Find the ratio of the sides. Put the sides opposite the 45◦ angles over each other, the sides opposite the 60◦ angles over each other, and the sides opposite the third angles over each other. What happens? AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar. The AA Similarity Postulate is a shortcut for showing that two triangles are similar. If two angles in one triangle are congruent to two angles in another, this is now enough information to show that the two triangles are similar. Then, use the similarity to ﬁnd the lengths of corresponding sides as needed. Example 1: Determine if the following two triangles are similar. If so, write the similarity statement.

Solution: Find the measure of the third angle in each triangle. m G = 48◦ and m M = 30◦ by the Triangle Sum Theorem. Therefore, all three angles are congruent, so the two triangles are similar. FEG ∼ MLN. Example 2: Determine if the following two triangles are similar. If so, write the similarity statement. Solution: m C = 39◦ and m F = 59◦ . The angles are not equal, ABC and DEF are not similar. 426

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Chapter 8. Similarity

Example 3: Are the following triangles similar? If so, write the similarity statement. Solution: Because AE || CD, A ∼ = D and C ∼ = E by the Alternate Interior Angles Theorem. Therefore, by the AA Similarity Postulate, ABE ∼ DBC.

Indirect Measurement

An application of similar triangles is to measure lengths indirectly. The length to be measured would be some feature that was not easily accessible to a person, such as: the width of a river or canyon or the height of a tall object. To measure something indirectly, you generally set up a pair of similar triangles. Example 4: A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If Ellie is 4 feet 10 inches tall, how tall is the tree? Solution: Draw a picture. From the diagram, the tree and Ellie are parallel, therefore the two triangles are similar to each other. Write a proportion.

5.5 f t 4 f t, 10in = xft 125 f t 427

8.3. Similarity by AA

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Notice that our measurements are not all in the same units. Change both numerators to inches and then cross multiply. 66in 58in = −→ 58(125) = 66(x) xft 125 f t 7250 = 66x x ≈ 109.848 f t Know What? Revisited It is safe to assume that George and the ﬂagpole stand vertically, making right angles with the ground. Also, the angle where the sun’s rays hit the ground is the same for both. The two triangles are similar. Set up a proportion. 10 6 = −→ 10x = 510 85 x x = 51 f t. The height of the ﬂagpole is 51 feet. Review Questions

Use the diagram to complete each statement.

1. SAM ∼ SM ? 2. SA ? = ? = RI 3. SM = 4. T R = 5. 9? = 8? Answer questions 6-9 about trapezoid ABCD.

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Chapter 8. Similarity

Name two similar triangles. How do you know they are similar? Write a true proportion. Name two other triangles that might not be similar. If AB = 10, AE = 7, and DC = 22, ﬁnd AC. Be careful! Writing How many angles need to be congruent to show that two triangles are similar? Why? Writing How do congruent triangles and similar triangles differ? How are they the same?

Use the triangles to the left for questions 12-15. AB = 20, DE = 15, and BC = k.

12. 13. 14. 15.

Are the two triangles similar? How do you know? Write an expression for FE in terms of k. If FE = 12, what is k? Fill in the blanks: If an acute angle of a _______ triangle is congruent to an acute angle in another ________triangle, then the two triangles are _______.

Determine whether the following triangles are similar. If so, write a similarity statement.

16.

17. 429

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18.

In order to estimate the width of a river, the following technique can be used. Use the diagram.

Place three markers, O,C, and E on the upper bank of the river. E is on the edge of the river and OC⊥CE. Go across the river and place a marker, N so that it is collinear with C and E. Then, walk along the lower bank of the river and place marker A, so that CN⊥NA. OC = 50 f eet,CE = 30 f eet, NA = 80 f eet. Is OC || NA? How do you know? Is OCE ∼ ANE? How do you know? What is the width of the river? Find EN. Can we ﬁnd EA? If so, ﬁnd it. If not, explain. Janet wants to measure the height of her apartment building. She places a pocket mirror on the ground 20 ft from the building and steps back until she can see the top of the build in the mirror. She is 18 in from the mirror and her eyes are 5 ft 3 in above the ground. The angle formed by her line of sight and the ground is congruent to the angle formed by the reﬂection of the building and the ground. Draw a diagram to illustrate this problem. How tall is the building? 24. Sebastian is curious to know how tall the announcers box is on his schools football ﬁeld. On a sunny day he measures the shadow of the box to be 45 ft and his own shadow is 9 ft. Sebastian is 5 ft 10 in tall. Find the height of the box. 25. Juanita spots a ship in the harbor and wonders how tall the mast is. The deck of the ship is the same height as the pier on which she is standing. The shadow of the mast is on the pier and she measures it to be 18 ft long. Juanita is 5 ft 4 in tall and her shadow is 4 ft long. How tall is the ship’s mast? 26. Use shadows or a mirror to measure the height of an object in your yard or on the school grounds. Draw a picture to illustrate your method.

19. 20. 21. 22. 23.

Use the diagram below to answer questions 27-31. 430

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Chapter 8. Similarity

27. Draw the three separate triangles in the diagram. 28. Explain why GDE ∼ = DFE ∼ = GFD. Complete the following proportionality statements. 29. 30. 31.

GF DF GF GD GE DE

= = =

? FE ? GE DE ?

Review Queue Answers

1.

2.

(a) x = 52◦ , y = 80◦ w 18 15 (b) 20 = 15 25 25 = z 25w = 15(20) 25(18) = 15z 25w = 300 450 = 15z w = 12 30 = z (a) 168 = 8x 6y = 216 x = 21 y = 36 21 (b) Answers will vary. One possibility: 28 8 = 6 (c) 28(12) = 8(6 + x) 336 = 48 + 8x 288 = 8x 36 = x Because x = 21, like in part a, this is not a true proportion.

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8.4. Similarity by SSS and SAS

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8.4 Similarity by SSS and SAS Learning Objectives

• Use SSS and SAS to determine whether triangles are similar. • Apply SSS and SAS to solve problems about similar triangles. Review Queue

1.

(a) What are the congruent angles? List each pair.

(b) Write the similarity statement. (c) If AB = 8, BD = 20, and BC = 25, ﬁnd BE. 2. Solve the following proportions. (a) (b) (c)

6 21 8 = x x+2 2x−1 6 = 15 x−3 4 9 = x+2

Know What? Recall from an earlier chapter that the game of pool relies heavily on angles; m 1 = m 2.

The dimensions of a pool table are 92 inches by 46 inches. You decide to hit the cue ball so it follows the yellow path shown. The horizontal and vertical distances are in the picture. Are the two triangles similar? Why? How far did the cue ball travel to get to the red ball? Link for an interactive game of pool: http://www.coolmath-games.com/0-poolgeometry/index.html 432

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Chapter 8. Similarity

SSS for Similar Triangles

If you do not know any angle measures, can you say two triangles are similar? Recall Investigation 4-2, Constructing a Triangle, given three sides. Investigation 8-2: SSS Similarity Tools Needed: ruler, compass, protractor, paper, pencil 1. Using Investigation 4-2, construct a triangle with sides 6 cm, 8 cm, and 10 cm.

2. Construct a second triangle with sides 9 cm, 12 cm, and 15 cm. 3. Using your protractor, measure the angles in both triangles. What do you notice? 4. Line up the corresponding sides. Write down the ratios of these sides. What happens?

To see an animated construction of this, click: http://www.mathsisfun.com/geometry/construct-ruler-compass-1.htm l From Step 3, notice that the angles in the two triangles are equal. Second, when the corresponding sides are lined 8 up, the sides are all in the same proportion, 69 = 12 = 10 15 . If you were to repeat this activity, for a 3-4-5 or 12-16-20 triangle, you will notice that they are all similar. That is because each of these triangles have sides that are multiples of 3-4-5. If we generalize the results of this investigation, the SSS Similarity Theorem emerges. SSS Similarity Theorem: If all the corresponding sides of two triangles are proportional, then the two triangles are similar. Example 1: Determine if any of the triangles below are similar.

Solution: Compare two triangles at a time. In the proportions, place the shortest sides over each other, the longest sides over each other, and the middle sides over each other. Then, determine if the proportions are equal. ABC and DEF:

20 15

=

22 16

=

24 18

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Reduce each fraction to see if they are equal. not similar. DEF and GHI: 15 30

= 12 , 16 33 =

16 33 ,

=

16 33

=

= 43 , 22 16 =

11 8,

and

24 18

= 43 . Because

4 3

=

11 8 , ABC

and DEF are

18 36

18 1 1 16 36 = 2 . Because 2 = 33 , DEF is not similar to GHI. 20 22 24 30 = 33 = 36 24 2 2 36 = 3 . Because all three ratios reduce to 3 , ABC ∼ GIH.

and

ABC and GHI: 20 30

15 30

20 15

2 = 23 , 22 33 = 3 , and

Example 2: Algebra Connection Find x and y, such that ABC ∼ DEF.

Solution: According to the similarity statement, the corresponding sides are: values:

AB DE

=

BC EF

=

AC DF .

Substituting known

9 4x − 1 18 = = 6 10 y 9 4x − 1 = 6 10 9(10) = 6(4x − 1)

9 18 = 6 y 9y = 18(6)

90 = 24x − 6

9y = 108

96 = 24x

y = 12

x=4 SAS for Similar Triangles

SAS is the last way to show two triangles are similar. If two sides are proportional AND the included angles are congruent, then the two triangles are similar. For the following investigation, use Investigation 4-3, Constructing a Triangle with SAS. Investigation 8-3: SAS Similarity Tools Needed: paper, pencil, ruler, protractor, compass 1. Using Investigation 4-3, construct a triangle with sides 6 cm and 4 cm and the included angle is 45◦ .

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2. Repeat Step 1 and construct another triangle with sides 12 cm and 8 cm and the included angle is 45◦ . 3. Measure the other two angles in both triangles. What do you notice?

4. Measure the third side in each triangle. Make a ratio. Is this ratio the same as the ratios of the sides you were given? SAS Similarity Theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in the ﬁrst triangle is congruent to the included angle in the second, then the two triangles are similar. In other words, If

AB XY

=

AC XZ

and A ∼ = X, then ABC ∼ XY Z.

Example 3: Are the two triangles similar? How do you know?

Solution: B ∼ = Z because they are both right angles. Second, AB BC = and ABC ∼ XZY . XZ ZY

10 15

=

24 36

because they both reduce to 23 . Therefore,

Notice with this example that we can ﬁnd the third sides of each triangle using the Pythagorean Theorem. If we were 2 to ﬁnd the third sides, AC = 39 and XY = 26. The ratio of these sides is 26 39 = 3 .

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Example 4: Are there any similar triangles? How do you know? Solution: A is shared by EAB and DAC, so it is congruent to itself. If two triangles would be similar.

AE AD

=

AB AC

then, by SAS Similarity, the

12 9 = 9 + 3 12 + 5 3 12 9 = = 12 4 17 Because the proportion is not equal, the two triangles are not similar. Example 5: From Example 4, what should BC equal for EAB ∼ DAC? 9 Solution: The proportion we ended up with was 12 = 34 = 12 17 . AC needs to equal 16, so that AC = AB + BC and 16 = 12 + BC. BC should equal 4 in order for EAB ∼ DAC.

12 16

= 34 . Therefore,

Similar Triangles Summary

To summarize, two triangles are similar if and only if:

TABLE 8.2: Name AA

Description Two angles in one triangle are congruent to two angles in another triangle.

SSS for Similar Triangles

All three sides in one triangle are proportional to three sides in another triangle.

SAS for Similar Triangles

Two sides in one triangle are proportional with two sides in another triangle AND the included angles are congruent.

Picture

Know What? Revisited First, ﬁnd the vertical length of the larger triangle. The two triangles are similar by AA, two right angles and 1 ∼ = 2. Set up a proportion.

17.5 29 = 23 v

Doing cross-multiplication, v = 38.1. Second, to ﬁnd the distance that the cue ball travels, use the Pythagorean Theorem. 17.52 + 232 = d12 and 38.12 + 292 = d22 , the lengths 28.9 and 47.9, and the total length is 76.8 inches. 436

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Review Questions

Use the following diagram for questions 1-3. The diagram is to scale.

1. 2. 3. 4.

Are the two triangles similar? Explain your answer. Are the two triangles congruent? Explain your answer. What is the scale factor for the two triangles? Writing How come there is no ASA Similarity Theorem?

Fill in the blanks in the statements below. Use the diagram to the left.

5. ABC ∼ BC AC 6. AB ? = ? = ? 7. If ABC had an altitude, AG = 10, what would be the length of altitude DH?

Use the diagram for questions 8-12. 437

8.4. Similarity by SSS and SAS

8. 9. 10. 11. 12.

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ABC ∼ Why are the two triangles similar? Find ED. BD ? DE ? = BC = ? CE Is AD DB = EB a valid proportion? How do you know?

Determine if the following triangles are similar. If so, write the similarity theorem and statement.

13.

14.

15.

16. 438

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17.

18. Algebra Connection Find the value of the missing variable(s) that makes the two triangles similar.

19.

20. 21. At a certain time of day, a building casts a 25 ft shadow. At the same time of day, a 6 ft tall stop sign casts a 15 ft shadow. How tall is the building? 22. A child who is 42 inches tall is standing next to the stop sign in #21. How long is her shadow? 23. An architect wants to build 3 similar right triangles such that the ratio of the middle triangle to the small triangle is the same as the ratio of the largest triangle to the middle triangle. The smallest one has side lengths 5, 12 and 13. The largest triangle has side lengths 45, 108 and 117. What are the lengths of the sides of the middle triangle? 24. Jaime wants to ﬁnd the height of a radio tower in his neighborhood. He places a mirror on the ground 30 ft from the tower and walks backwards 3 ft until he can see the top of the tower in the mirror. Jaime is 5 ft 6 in tall. How tall is the radio tower? For questions 25-27, use ABC with A(−3, 0), B(−1.5, 3) and C(0, 0) and DEF with D(0, 2), E(1, 4) and F(2, 2). 25. Find AB, BC, AC, DE, EF and DF. 26. Use these values to ﬁnd the following proportions: 27. Are these triangles similar? Justify your answer.

AB BC DE , EF

and

AC DF .

For questions 28-31, use CAR with C(−3, 3), A(−3, −1) and R(0, −1) and LOT with L(5, −2), O(5, 6) and T (−1, 6). 439

8.4. Similarity by SSS and SAS 28. 29. 30. 31.

Find the slopes of CA, AR, LO and OT . What are the measures of A and O? Explain. Find LO, OT,CA and AR. Use these values to write the ratios LO : CA and OT : AR. Are the triangles similar? Justify your answer.

Review Queue Answer

(a) A ∼ = D, E ∼ = C (b) ABE ∼ DBC (c) BE = 10 2. (a) 68 = 21 x , x = 28 (b) 15(x + 2) = 6(2x − 1) 15x + 30 = 12x − 6 3x = −36 x = −12 3. (x − 3)(x + 2) = 36 x2 − x − 6 = 36 2 x − x − 42 = 0 (x − 7)(x + 6) = 0 x = 7, −6 1.

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8.5 Proportionality Relationships Learning Objectives

• Identify proportional segments when two sides of a triangle are cut by a segment parallel to the third side. • Extend triangle proportionality to parallel lines.

Review Queue

1. Write a similarity statement for the two triangles in the diagram. Why are they similar?

2. 3. 4. 5.

If XA = 16, XY = 18, XB = 32, ﬁnd XZ. If Y Z = 27, ﬁnd AB. Find AY and BZ. AY BZ = BX ? Is AX

Know What? To the right is a street map of part of Washington DC. R Street, Q Street, and O Street are parallel and 7th Street is perpendicular to all three. R and Q are one city block (usually 14 mile or 1320 feet) apart. The other given measurements are on the map. What are x and y?

What is the distance from: • R and 7th to R and Florida? • Q and 7th to Q and Florida? • O and 7th to O and Florida? 441

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Triangle Proportionality

Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides proportionally. Example 1: A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?

Solution: The midsegments endpoints are the midpoints of the two sides it connects. The midpoints split the sides evenly. Therefore, the ratio would be a : a or b : b. Both of these reduce to 1:1. The midsegment divides the two sides of the triangle proportionally, but what about other segments? Investigation 8-4: Triangle Proportionality Tools Needed: pencil, paper, ruler 1. Draw ABC. Label the vertices. 2. Draw XY so that X is on AB and Y is on BC. X and Y can be anywhere on these sides.

YC 3. Is XBY ∼ ABC? Why or why not? Find AX, XB, BY, and YC. Then set up the ratios AX XB and Y B . Are they equal? 4. Draw a second triangle, DEF. Label the vertices. 5. Draw XY so that X is on DE and Y is on EF AND XY || DF. FY 6. Is XEY ∼ DEF? Why or why not? Find DX, XE, EY, and Y F. Then set up the ratios DX XE and Y E . Are they equal?

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From this investigation, it is clear that if the line segments are parallel, then XY divides the sides proportionally. Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Proof of the Triangle Proportionality Theorem

Given: ABC with DE || AC Prove:

AD DB

= CE EB

TABLE 8.3: Statement 1. DE || AC 2. 1 ∼ = 2, 3 ∼ = 4 3. ABC ∼ DBE 4. AD + DB = AB EC + EB = BC AB BC = BE 5. BD 6. 7. 8.

AD+DB = EC+EB BD BE AD DB EC BE + = BD DB BE + BE AD EC BD + 1 = BE + 1

9.

AD BD

=

EC BE

Reason Given Corresponding Angles Postulate AA Similarity Postulate Segment Addition Postulate Corresponding sides in similar triangles are proportional Substitution PoE Separate the fractions Substitution PoE (something over itself always equals 1) Subtraction PoE

Although the proof of the converse of this theorem will not be shown, it is essentially the same proof in the reverse BE order. Using the corollaries from earlier in this chapter, BD DA = EC is also a true proportion. Example 2: In the diagram below, EB || BD. Find BC.

Solution: Use the Triangle Proportionality Theorem. 443

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10 BC = −→ 15(BC) = 120 15 12 BC = 8 Example 3: Is DE || CB?

Solution: Use the Triangle Proportionality Converse. If the ratios are equal, then the lines are parallel. 6 18

=

1 3

and

8 24

=

1 3

Because the ratios are equal, DE || CB. Parallel Lines and Transversals

We can extend the Triangle Proportionality Theorem to multiple parallel lines. Theorem 8-7: If three parallel lines are cut by two transversals, then they divide the transversals proportionally. Example 4: Find a.

Solution: The three lines are marked parallel, so set up a proportion.

9 a = 20 15 180 = 15a a = 12 Theorem 8-7 can be expanded to any number of parallel lines with any number of transversals. When this happens all corresponding segments of the transversals are proportional. Example 5: Find a, b, and c. 444

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Chapter 8. Similarity

Solution: Look at the corresponding segments. Only the segment marked 2 is opposite a number, all the other segments are opposite variables. That means we will be using this ratio, 2:3 in all of our proportions.

a 9 = 2 3 3a = 18

2 3 = 4 b 2b = 12

a=6

b=6

2 3 = 3 c 2c = 9 c = 4.5

There are several ratios you can use to solve this example. To solve for b, you could have used the proportion which will still give you the same answer.

6 4

= 9b ,

Proportions with Angle Bisectors

The last proportional relationship we will explore is how an angle bisector intersects the opposite side of a triangle. − → BC AB = AD . The proof is By deﬁnition, AC divides BAD equally, so BAC ∼ = CAD. The proportional relationship is CD in the review exercises. Theorem 8-8: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides. Example 6: Find x.

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Solution: Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:

9 21 = x 14 21x = 126 x=6 Example 7: Algebra Connection Determine the value of x that would make the proportion true.

Solution: You can set up this proportion just like the previous example. 5 4x + 1 = 3 15 75 = 3(4x + 1) 75 = 12x + 3 72 = 12x 6=x Know What? Revisited To ﬁnd x and y, set up a proportion using parallel the parallel lines.

2640 1320 1980 = = x 2380 y From this, x = 4760 f t and y = 3570 f t. To ﬁnd a, b, and c, use the Pythagorean Theorem.

26402 + a2 = 47602 39602 + b2 = 71402 59402 + c2 = 107102

a = 3960.808, b = 5941.212, c = 8911.818 446

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Review Questions

Use the diagram to answers questions 1-5. DB || FE.

1. Name the similar triangles. Write the similarity statement. BE ? = FC 2. EC EC 3. CB = CF ? DB BC 4. ? = EC ? 5. FC+? FC = FE Use the diagram to answer questions 6-10. AB || DE.

6. 7. 8. 9. 10.

Find BD. Find DC. Find DE. Find AC. We know that

BD DC

=

AE EC

and

BA DE

=

BC DC .

Why is

BA DE

=

BD DC ?

Use the given lengths to determine if AB || DE. 447

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11.

12.

13.

14.

Algebra Connection Find the value of the missing variable(s).

15.

16. 448

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Chapter 8. Similarity

17.

18.

19.

20.

Find the value of each variable in the pictures below.

21. 449

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22.

23.

24.

Find the unknown lengths.

25.

26. 27. Error Analysis 450

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Chapter 8. Similarity

Casey attempts to solve for a in the diagram using the proportion 5 6 = a 5 What did Casey do wrong? Write the correct proportion and solve for a. 28. Michael has a triangular shaped garden with sides of length 3, 5 and 6 meters. He wishes to make a path along the perpendicular bisector of the angle between the sides of length 3 m and 5 m. Where will the path intersect the third side?

29. This is a map of lake front properties. Find a and b, the length of the edge of Lot 1 and Lot 2 that is adjacent to the lake. 30. Fill in the blanks of the proof of Theorem 8-8.

− → −→ ← → Given: BAD with AC is the angle bisector of BAD Auxiliary lines AX and XD, such that X, A, B are − → ← → collinear and AC || XD. BC BA Prove: CD = AD

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TABLE 8.4: Statement − → 1. AC is the angle bisector of BADX, A, B are collinear − → ← → and AC || XD 2. BAC ∼ = CAD 3. 4. CAD ∼ = ADX ∼ 5. X = ADX 6. XAD is isosceles 7. 8. 9. 10. Review Queue Answers

1. AXB ∼ Y XZ by AA Similarity Postulate 32 2. 16 18 = XZ , XZ = 36 AB 3. 16 18 = 27 , AB = 24 4. AY = 18 − 16 = 2, BZ = 36 − 32 = 4 2 4 = 32 . Yes, this is a true proportion. 5. 16

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Reason

Corresponding Angles Postulate

Deﬁnition of an Isosceles Triangle Congruent segments are also equal If three parallel lines are cut by two transversals, then they divide the transversals proportionally

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8.6 Similarity Transformations Learning Objectives

• Draw a dilation of a given ﬁgure. • Plot an image when given the center of dilation and scale factor. • Determine if one ﬁgure is the dilation of another. Review Queue

1. Are the two quadrilaterals similar? How do you know?

2. What is the scale factor from XY ZW to CDAB? Leave as a fraction. 3. Quadrilateral EFGH has vertices E(−4, −2), F(2, 8), G(6, 2) and H(0, −4). Quadrilateral LMNO has vertices L(−2, −1), M(1, 4), N(3, 1), and O(0, −2). Determine if the two quadrilaterals are similar. Explain your reasoning. Know What? One practical application of dilations is perspective drawings. These drawings use a vanishing point (the point where the road meets the horizon) to trick the eye into thinking the picture is three-dimensional. The picture to the right is a one-point perspective and is typically used to draw streets, train tracks, rivers or anything else that is linear.

There are also two-point perspective drawings, which are very often used to draw a street corner or a scale drawing of a building. Both of these drawing are simple representations of one and two perspective drawings. Your task for this Know What? is to draw your own perspective drawing with either one or two vanishing points and at least 5 objects. Each object should have detail (windows, doors, sign, stairs, etc.) 453

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Dilations

A dilation makes a ﬁgure larger or smaller, but has the same shape as the original. In other words, a dilation is similar to the original. Transformation: An operation that moves, ﬂips, or changes a ﬁgure to create a new ﬁgure. Transformations that preserve size are rigid and ones that do not are non-rigid. Dilation: A non-rigid transformation that preserves shape but not size. All dilations have a center and a scale factor. The center is the point of reference for the dilation (like the vanishing point in a perspective drawing) and scale factor tells us how much the ﬁgure stretches or shrinks. A scale factor is typically labeled k and is always greater than zero. Also, if the original ﬁgure is labeled ABC, for example, the dilation would be A B C . Recall that the tick mark ("prime") indicates that it is a copy. A second dilation would be A”, or "A double-prime." Example 1: The center of dilation is P and the scale factor is 3. Find Q .

Solution: If the scale factor is 3 and Q is 6 units away from P, then Q is going to be 6 × 3 = 18 units away from P. Because we are only dilating apoint, the dilation will be collinear with the original and center.

Example 2: Using the picture above, change the scale factor to 13 . Find Q

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Chapter 8. Similarity

Solution: Now the scale factor is 13 , so Q

is going to be 13 the distance away from P as Q is. In other words, Q

is going to be 6 × 13 = 2 units away from P. Q

will also be collinear with Q and center. Example 3: KLMN is a rectangle with length 12 and width 8. If the center of dilation is K with a scale factor of 2, draw K’L’M’N’.

Solution: If K is the center of dilation, then K and K’ will be the same point. From there, L’ will be 8 units above L and N’ will be 12 units to the right of N.

Example 4: Find the perimeters of KLMN and K’L’M’N’. Compare this to the scale factor. Solution: The perimeter of KLMN = 12 + 8 + 12 + 8 = 40. The perimeter of K L M N = 24 + 16 + 24 + 16 = 80. The ratio of the perimeters is 80:40 or 2:1, which is the same as the scale factor. Example 5: ABC is a dilation of DEF. If P is the center of dilation, what is the scale factor?

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Solution: Because ABC is a dilation of DEF, we know that the triangles are similar. Therefore the scale factor is the ratio of the sides. Since ABC is smaller than the original, DEF, the scale factor is going to be a fraction 3 less than one, 12 20 = 5 . If DEF was the dilated image, the scale factor would have been 53 . If the dilated image is smaller than the original, then the scale factor is 0 < k < 1. If the dilated image is larger than the original, then the scale factor is k > 1. Dilations in the Coordinate Plane

In this text, the center of dilation will always be the origin, unless otherwise stated. Example 6: Determine the coordinates of ABC and A B C and ﬁnd the scale factor.

Solution: The coordinates of ABC are A(2, 1), B(5, 1) and C(3, 6). The coordinates of A B C are A(6, 3), B(15, 3) and C(9, 18). By looking at the corresponding coordinates, each is three times the original. That means k = 3. Again, the center, original point, and dilated point are collinear. Therefore, you can draw a ray from the origin to C , B , and A such that the rays pass through C, B, and A, respectively. Lets show that dilations are a similarity transformation (preserves shape). Using the distance formula, we will ﬁnd the lengths of the sides of both triangles in Example 6 to demonstrate this. ABC √ AB = (2 − 5)2 + (1 − 1)2 = 9 = 3 √ AC = (2 − 3)2 + (1 − 6)2 = 26 √ CB = (3 − 5)2 + (6 − 1)2 = 29

A B C

√ A B = (6 − 15)2 + (3 − 3)2 = 81 = 9 √ √ A C = (6 − 9)2 + (3 − 18)2 = 234 = 3 26 √ √ C B = (9 − 15)2 + (18 − 3)2 = 261 = 3 29

From this, we also see that all the sides of A B C are three times larger than ABC. Therefore, a dilation will always produce a similar shape to the original. In the coordinate plane, we say that A is a mapping of A. So, if the scale factor is 3, then A(2, 1) is mapped to (usually drawn with an arrow) A (6, 3). The entire mapping of ABC can be written (x, y) → (3x, 3y) because k = 3. For any dilation the mapping will be (x, y) → (kx, ky). 456

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Know What? Revisited Answers to this project will vary depending on what you decide to draw. Make sure that you have at least ﬁve objects with some sort of detail. If you are having trouble getting started, go to the website: http://www.drawing-and-painting-techniques.com/drawing-perspective.html

Review Questions

Given A and the scale factor, determine the coordinates of the dilated point, A . You may assume the center of dilation is the origin. 1. A(3, 9), k = 23 2. A(−4, 6), k = 2 3. A(9, −13), k = 12 Given A and A , ﬁnd the scale factor. You may assume the center of dilation is the origin. 4. A(8, 2), A (12, 3) 5. A(−5, −9), A (−45, −81) 6. A(22, −7), A(11, −3.5) In the two questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black ﬁgure is the original and P is the center of dilation. 7. k = 4

8. k =

1 3

In the two questions below, ﬁnd the scale factor, given the corresponding sides. In each diagram, the black ﬁgure is the original and P is the center of dilation. 457

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9.

10. 11. Find the perimeter of both triangles in #7. What is the ratio of the perimeters? 12. Writing What happens if k = 1? The origin is the center of dilation. Find the coordinates of the dilation of each ﬁgure, given the scale factor. 13. A(2, 4), B(−3, 7),C(−1, −2); k = 3 14. A(12, 8), B(−4, −16),C(0, 10); k =

3 4

Multi-Step Problem Questions 15-21 build upon each other. 15. Plot A(1, 2), B(12, 4),C(10, 10). Connect to form a triangle. 16. Make the origin the center of dilation. Draw 4 rays from the origin to each point from #15. Then, plot A (2, 4), B (24, 8),C (20, 20). What is the scale factor? 17. Use k = 4, to ﬁnd A

B

C

. Plot these points. 18. What is the scale factor from A B C to A

B

C

? 19. Find (Ois the origin): (a) (b) (c) (d) (e)

OA AA

AA

OA

OA

20. Find: (a) AB (b) A B

(c) A

B

21. Compare the ratios: (a) OA : OA and AB : A B

(b) OA : OA

and AB : A

B

Algebra Connection For questions 22-27, use quadrilateral ABCD with A(1, 5), B(2, 6),C(3, 3) and D(1, 3) and its transformation A B C D with A (−3, 1), B (0, 4),C (3, −5) and D (−3, −5). 458

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Chapter 8. Similarity

Plot the two quadrilaterals in the coordinate plane. ←→ Find the equation of CC . ←→ Find the equation of DD . Find the intersection of these two lines algebraically or graphically. What is the signiﬁcance of this point? What is the scale factor of the dilation?

Construction Use a compass and straight edge to construct a dilation as well. Copy the diagram below.

28. Set a compass to CG and use this setting to mark off a point 3 times as far from C as G is. Label this point G . Repeat this process for CO and CD to ﬁnd O and D .

O O G

D

, OG and GGD . 29. Connect G , O and D to make D O G . Find the ratios, DDO 30. What is the scale factor of this dilation? 31. Describe how you would dilate the ﬁgure by a scale factor of 4. 32. Describe how you would dilate the ﬁgure by a scale factor of 12 . Review Queue Answers

1. Yes, all the angles are congruent and the corresponding sides are in the same ratio. 2. 53 3. Yes, LMNO ∼ EFGH because LMNO is exactly half of EFGH.

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8.7 Extension: Self-Similarity Learning Objectives

• Draw sets of the Sierpinski Triangle. • Understand basic fractals. Self-Similar: When one part of an object can be enlarged (or shrunk) to look like the whole object. To explore self-similarity, we will go through a couple of examples. Typically, each step of repetition is called an iteration or level. The ﬁrst level is called the Start Level or Stage 0. Sierpinski Triangle

The Sierpinski triangle iterates an equilateral triangle (but, any triangle can be used) by connecting the midpoints of the sides and shading the central triangle (Stage 1). Repeat this process for the unshaded triangles in Stage 1 to get Stage 2. This series was part of the Know What? in Section 5.1.

Example 1: Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangles. Determine if there is a pattern. Solution:

TABLE 8.5: Unshaded Shaded

Stage 0 1 0

Stage 1 3 1

Stage 2 9 4

Stage 3 27 13

The unshaded triangles seem to be powers of 3, 30 , 31 , 32 , 33 , . . . The shaded triangles are add the previous number of unshaded triangles to the total. For Example, Stage 4 would equal 9 + 13 shaded trangles.

Fractals

A fractal is another self-similar object that is repeated at successively smaller scales. Below are the ﬁrst three stages of the Koch snowﬂake. 460

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Chapter 8. Similarity

Example 2: Determine the number of edges and the perimeter of each snowﬂake.

TABLE 8.6: Number of Edges Edge Length Perimeter

Stage 0 3 1 3

Stage 1 12

Stage 2 48

1 3

1 9 48 9

4

= 5.3

The Cantor Set

The Cantor set is another fractal that consists of dividing a segment into thirds and then erasing the middle third.

Review Questions

1. Draw Stage 4 of the Cantor set. 2. Use the Cantor Set to ﬁll in the table below.

TABLE 8.7:

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

3. 4. 5. 6. 7.

Number of Segments

Length of each Segment

1 2 4

1

Total Length of the Segments 1

1 3 1 9

2 3 4 9

How many segments are in Stage n? What is the length of each segment in Stage n? Draw Stage 3 of the Koch snowﬂake. Fill in the table from Example 2 for Stage 3 of the Koch snowﬂake. A variation on the Sierpinski triangle is the Sierpinski carpet, which splits a square into 9 equal squares, coloring the middle one only. Then, split the uncolored squares to get the next stage. Draw the ﬁrst 3 stages of this fractal. 461

8.7. Extension: Self-Similarity

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8. How many shaded vs. unshaded squares are in each stage? 9. Fractals are very common in nature. For example, a fern leaf is a fractal. As the leaves get closer to the end, they get smaller and smaller. Find three other examples of fractals in nature.

10. Use the internet to explore fractals further. Write a paragraph about another example of a fractal in music, art or another ﬁeld that interests you.

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Chapter 8. Similarity

8.8 Chapter 8 Review Keywords and Theorems

• • • • • • • • • • • • • • • • • • • • • • • • • • • •

Ratio Proportion Means Extremes Cross-Multiplication Theorem Corollary Corollary 8-1 Corollary 8-2 Corollary 8-3 Corollary 8-4 Corollary 8-5 Similar Polygons Scale Factor Theorem 8-2 AA Similarity Postulate Indirect Measurement SSS Similarity Theorem SAS Similarity Theorem Triangle Proportionality Theorem Triangle Proportionality Theorem Converse Theorem 8-7 Theorem 8-8 Transformation Rigid Transformation Non-rigid Transformation Dilation Self-Similar Fractal

Review Questions

1. Solve the following proportions. (a) (b)

x+3 10 3 = 2 8 2x−1 5 = x+3

2. The extended ratio of the angle in a triangle are 5:6:7. What is the measure of each angle? 3. Rewrite 15 quarts in terms of gallons.

Determine if the following pairs of polygons are similar. If it is two triangles, write why they are similar. 463

8.8. Chapter 8 Review

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4.

5.

6.

7.

8.

9. 10. Draw a dilation of A(7, 2), B(4, 9), and C(−1, 4) with k = 32 .

Algebra Connection Find the value of the missing variable(s). 464

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Chapter 8. Similarity

11.

12.

13.

14. Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/ﬂexr/chapter/9692 .

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